The poll shows Romney at 51 percent among a sample of likely voters in the 12 states, while Obama is at 46 percent.

Perhaps most strikingly, the poll shows Romney running even with Obama among women, with the two candidates tied at 48 percent.

The poll comes a week after most swing state polling showed Obama holding relatively steady. Republicans expecting a significant bounce from Romney’s debate performance two weeks ago had yet to see that momentum in the states that matter most.

The Gallup poll appears to be the first major poll covering multiple states to show that momentum registering in the states that will decide the presidency.

The same poll in mid-September showed Obama at 48 percent and Romney at 46 percent in the swing states. That poll, though, was conducted among registered voters, while the new poll is conducted among likely voters — a higher standard and one that reflects more enthusiastic voters.

(Indeed, the current poll shows the race is largely unchanged when it comes to registered voters, with Obama at 49 percent and Romney at 47 percent.)

Republicans cited the poll as a sign of their progress, while Democrats suggested the likely voter model was flawed.

Obama’s campaign released an entire memo casting doubt on the poll.

“The latest Gallup/USA Today Battleground survey showing President Obama and Governor Romney tied with women in battleground states (48-48) is an extreme outlier, defying the trends seen in every other battleground and national poll,” Obama pollster Joel Benenson said.